TY - JOUR
T1 - A Positivity-Preserving, Energy Stable And Convergent Numerical Scheme For The Poisson-Nernst-Planck System
AU - Liu, Chun
AU - Wang, Cheng
AU - Wise, Steven M.
AU - Yue, Xingye
AU - Zhou, Shenggao
N1 - Funding Information:
Received by the editor September 11, 2020, and, in revised form, December 27, 2020. 2020 Mathematics Subject Classification. Primary 35K35, 35K55, 49J40, 65M06, 65M12. Key words and phrases. Poisson-Nernst-Planck (PNP) system, logarithmic energy potential, positivity preserving, energy stability, optimal rate convergence analysis, higher order asymptotic expansion. This work was supported in part by the National Science Foundation (USA) grants NSF DMS-1759535, NSF DMS-1759536 (first author), NSF DMS-2012669 (second author), NSF DMS-1719854, DMS-2012634 (third author), National Natural Science Foundation of China 11971342 (fourth author), NSFC 21773165, Natural Science Foundation of Jiangsu Province (BK20200098), China, Young Elite Scientist Sponsorship Program by Jiangsu Association for Science and Technology, and National Key R&D Program of China 2018YFB0204404 (fifth author). The fifth author is the corresponding author.
Publisher Copyright:
© 2021 American Mathematical Society
PY - 2021/9
Y1 - 2021/9
N2 - In this paper we propose and analyze a finite difference numerical scheme for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility H−1 gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivity-preserving property for both concentrations, n and p, is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of 0 prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish the l∞ bound for n and p), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this article, which demonstrates the robustness of the proposed numerical scheme.
AB - In this paper we propose and analyze a finite difference numerical scheme for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility H−1 gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivity-preserving property for both concentrations, n and p, is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of 0 prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish the l∞ bound for n and p), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this article, which demonstrates the robustness of the proposed numerical scheme.
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U2 - 10.1090/mcom/3642
DO - 10.1090/mcom/3642
M3 - Article
AN - SCOPUS:85102117804
VL - 90
SP - 2071
EP - 2106
JO - Mathematics of Computation
JF - Mathematics of Computation
SN - 0025-5718
IS - 331
ER -